Question
Given the points \( (2,-7) \) and \( (0,4) \) on a line, find its equation in the form \( y=m x+b \). \( y=\square \)
Ask by Schultz Martin. in the United States
Feb 04,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The equation of the line is \( y = -\frac{11}{2}x + 4 \).
Solution
To find the equation of the line passing through the points \( (2, -7) \) and \( (0, 4) \) in the form \( y = mx + b \), follow these steps:
1. **Determine the Slope (\( m \))**:
The slope of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Plugging in the given points:
\[
m = \frac{4 - (-7)}{0 - 2} = \frac{11}{-2} = -\frac{11}{2}
\]
2. **Find the Y-Intercept (\( b \))**:
The y-intercept is the value of \( y \) when \( x = 0 \). From the point \( (0, 4) \), it's clear that:
\[
b = 4
\]
3. **Write the Equation**:
Substitute \( m \) and \( b \) into the slope-intercept form:
\[
y = mx + b
\]
\[
y = -\frac{11}{2}x + 4
\]
**Final Answer:**
\[
y = -\frac{11}{2}\,x + 4
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Extra Insights
To find the equation of the line in slope-intercept form \( y = mx + b \), we first need to calculate the slope \( m \) using the two points \( (2, -7) \) and \( (0, 4) \). The formula for slope is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - (-7)}{0 - 2} = \frac{4 + 7}{-2} = \frac{11}{-2} = -\frac{11}{2} \] Now, we can use one of the points, say \( (0, 4) \), to find the y-intercept \( b \). Since the x-coordinate is 0, we know the y-intercept is directly \( b = 4 \). Now we can write the equation in slope-intercept form: \[ y = -\frac{11}{2}x + 4 \] So, the equation of the line is: \( y = -\frac{11}{2}x + 4 \)
